Normal extension: Difference between revisions
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In [[algebra]], a '''normal extension''' of [[field (mathematics)|fields]] is a [[field extension]] ''E''/''F'' which contains all the roots of an irreducible polynomial if it contains one such root. | In [[algebra]], a '''normal extension''' of [[field (mathematics)|fields]] is a [[field extension]] ''E''/''F'' which contains all the roots of an irreducible polynomial if it contains one such root. | ||
A '''normal closure''' is a normal extension ''N''/''F'' with the property that no subfield of ''N'' is a normal extension of ''F''. Given any finite degree extension ''E''/''F'' there is a minimal finite degree normal extension ''N'' containing ''E'': this will be "the" normal closure of ''E'' over ''F''; any two normal closures are ''L''-isomorphic. | A '''normal closure''' is a normal extension ''N''/''F'' with the property that no subfield of ''N'' is a normal extension of ''F''. Given any finite degree extension ''E''/''F'' there is a minimal finite degree normal extension ''N'' containing ''E'': this will be "the" normal closure of ''E'' over ''F''; any two normal closures are ''L''-isomorphic.[[Category:Suggestion Bot Tag]] | ||
Latest revision as of 16:00, 26 September 2024
In algebra, a normal extension of fields is a field extension E/F which contains all the roots of an irreducible polynomial if it contains one such root.
A normal closure is a normal extension N/F with the property that no subfield of N is a normal extension of F. Given any finite degree extension E/F there is a minimal finite degree normal extension N containing E: this will be "the" normal closure of E over F; any two normal closures are L-isomorphic.